When testing for convergence, how do you determine which test to use?

1 Answer

There is no general method of determining the test you should use to check the convergence of a series.

  • For series where the general term has exponents of #n#, it's useful to use the root test (also known as Cauchy's test).
    Example 1: Power Series
    The definition of the convergence radius of the of a power series comes from the Cauchy test (however, the actual computation is usually done with the following test).

  • Generally, the computation of the ratio test (also known as d'Alebert's test) is easier than the computation of the root test.
    Example 2: Inverse Factorial
    For the series #sum_(n=1)^(oo) 1/(n!)# the d'Alembert's test gives us:
    #lim_(n to oo) |1/((n+1)!)|/|1/(n!)| = lim_(n to oo) |n!|/(|(n+1)!|) = lim_(n to oo) |(n!)/((n+1)n!)| = lim_(n to oo) |1/(n+1)| = 0#
    So the series is convergent.

  • If you know the result of the improper integral of the function #f(x)# such that #f(n)=a_n#, where #a_n# is the general term of the series being analyzed, then it might be a good idea to use the integral test.
    Example 3: A proof for the Harmonic Series.
    Knowing that the improper integral #int_1^(oo) 1/x dx# is divergent (it's easy to check) implies that the harmonic series #sum_(n=1)^(oo) 1/(n)# diverges.

  • Comparision tests are only useful if you know an appropriate series to compare the one you're analyzing to. However, they can be very powerful.
    Example 4: Hyperharmonic Series
    The series of the form #sum_(n=1)^(oo) 1/(n^p)# are called hyperharmonic series or #p#-series. If you can show that the series #sum_(n=1)^(oo) 1/(n^(1+epsilon))# converges, for some small, positive value of #epsilon#, than any #p#-series such that #p>1 + epsilon# converges.